**Victor Schlegel** (4 March 1843 – 22 November 1905)^{ [1] } was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams.

In the nineteenth century there were various expansions of the traditional field of geometry through the innovations of hyperbolic geometry, non-Euclidean geometry and algebraic geometry. Hermann Grassmann was one of the more advanced innovators with his anticipation of linear algebra and multilinear algebra that he called "Extension theory" (*Ausdehnungslehre*). As recounted by David E. Rowe in 2010:

- The most important new convert was Victor Schlegel, Grassmann’s colleague at Stettin Gymnasium from 1866 to 1868. Afterward Schlegel accepted a position as Oberlehrer at the Gymnasium in Waren, a small town in Mecklenburg.

In 1872 Schlegel published the first part of his *System der Raumlehre* which used Grassmann’s methods to develop plane geometry. Schlegel used his book to put forth Grassmann’s case, arguing that "Grassmann’s ideas had been neglected because he had not held a university chair." Continuing his criticism of academics, Schlegel expressed the reactionary view that

- Rather than developing a sound basis for their analytic methods, contemporary mathematicians tended to invent new symbolisms on an ad hoc basis, creating a Babel-like cacophony of unintelligible languages.

Schlegel’s attitude was that no basis for scientific method was shown in mathematics, and "neglect of foundations had led to the widely acknowledged lack of interest in mathematics in the schools."

The mathematician Felix Klein addressed Schlegel’s book in a review^{ [2] } criticizing him for neglect of cross ratio and failure to contextualize Grassmann in the flow of mathematical developments. Rowe indicates that Klein was most interested in developing his Erlangen program. In 1875 Schlegel countered with the second part of his *System der Raumlehre*, answering Klein in the preface. This part developed conic sections, harmonic ranges, projective geometry, and determinants.

Schlegel published a biography of Hermann Grassmann in 1878. Both parts of his textbook, and the biography, are now available at the Internet Archive; see External links section below.

At the Summer meeting of the American Mathematical Society on August 15, 1894, Schlegel presented an essay on the problem of finding the place which is at a minimum total distance from given points.^{ [3] }

In 1899 Schlegel became German national secretary for the international Quaternion Society and reported on it in Monatshefte für Mathematik. ^{ [4] }

**David Hilbert** was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics.

**Josip Plemelj** was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory. He was the first chancellor of the University of Ljubljana.

**Christian Felix Klein** was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time.

**Hermann Günther Grassmann** was a German polymath, known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties.

In mathematics, **multilinear algebra** extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of *p*-vectors and multivectors with Grassmann algebras.

**Friedrich Wilhelm Karl Ernst Schröder** was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic, by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce. He is best known for his monumental *Vorlesungen über die Algebra der Logik*, in three volumes, which prepared the way for the emergence of mathematical logic as a separate discipline in the twentieth century by systematizing the various systems of formal logic of the day.

**Eduard Study**, more properly **Christian Hugo Eduard Study**, was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.

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The **Quaternion Society** was a scientific society, self-described as an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the academic world that were experimenting with quaternions and other hypercomplex number systems. The group's guiding light was Alexander Macfarlane who served as its secretary initially, and became president in 1909. The association published a *Bibliography* in 1904 and a *Bulletin* from 1900 to 1913.

The **German Mathematical Society** is the main professional society of German mathematicians and represents German mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). It was founded in 1890 in Bremen with the set theorist Georg Cantor as first president. Founding members included Georg Cantor, Felix Klein, Walther von Dyck, David Hilbert, Hermann Minkowski, Carl Runge, Rudolf Sturm, Hermann Schubert, and Heinrich Weber.

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**Margarethe Kahn** was a German mathematician and Holocaust victim. She was among the first women to obtain a doctorate in Germany. Her doctoral work was on the topology of algebraic curves.

**Ioannis "John" N. Hazzidakis** was a Greek mathematician, known for the Hazzidakis transform in differential geometry. The Hazzidakis formula for the Hazzidakis transform can be applied in proving Hilbert's theorem on negative curvature, stating that hyperbolic geometry does not have a model in 3-dimensional Euclidean space.

**Carl Friedrich Geiser** was a Swiss mathematician, specializing in algebraic geometry. He is known for the Geiser involution and Geiser's minimal surface.

**Erhard Scholz** is a German historian of mathematics with interests in the history of mathematics in the 19th and 20th centuries, historical perspective on the philosophy of mathematics and science, and Hermann Weyl's geometrical methods applied to gravitational theory.

**Johann Jakob Burckhardt** was a Swiss mathematician and crystallographer. He was an invited speaker at the International Congress of Mathematicians in 1936 in Oslo.

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- ↑ Eneström, Gustaf Hjalmar (1905). "Nekrologe. Victor Schlegel (1843–1905)".
*Bibliotheca mathematica*. Leipzig: B. G. Teubner. p. 421. - ↑
*Jahrbuch über die Fortschritte der Mathematik*Jahrgang 1872, SS. 231–5 - ↑ Schlegel from Hagen in Westfalen (1894). "On the problem of the minimum sum of the distances of a point from given points".
*Bulletin of the American Mathematical Society*.**1**(2): 33–52. doi: 10.1090/s0002-9904-1894-00242-0 . MR 1557288. - ↑ Victor Schlegel (1899) "Internationaler Verein zur Beförderung des Studiums der Quaternionen und verwandter Systeme der Mathematik", Monatshefte für Mathematik 10(1):376

- David E. Rowe (2010) "Debating Grassmann’s Mathematics: Schlegel Versus Klein", Mathematical Intelligencer 32(1):41–8.
- Victor Schlegel (1883) "Theorie der homogen zusammengesetzten Raumgebilde",
*Nova Acta, Ksl. Leop.-Carol. Deutsche Akademie der Naturforscher*, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden. - Victor Schlegel (1886)
*Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper*, Waren. - Victor Schlegel (1896) "Ein Beitrag zur Geschichte der Mathematik in den letzten fünfzig Jahren",
*Zeitschrift für Mathematik und Physik*4;:1–21, 41–59.

- Schlegel (1878) Lehrbuch der elementaren Mathematik on the Internet Archive.
- Schlegel (1878) Hermann Grassmann: Sein Leben und seine Werke on the Internet Archive.
- Schlegel (1872) System der Raumlehre, part 1 on the Internet Archive.
- Schlegel (1875) System der Raumlehre, part 2 on the Internet Archive.

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